Applied Mathematics-A Lagrangian Scheme for the solution of nonlinear diffusion equations

Speaker: Bertram Düring (Department of Mathematics University of Sussex)

Date and Time: Thursday, December 7, 11am-12pm

Place: Elliott School (1957 E St. #214)

Title: A Lagrangian Scheme for the solution of nonlinear diffusion equations

Abstract: Nonlinear diffusion equations whose dynamics are driven by internal energies and given external potentials, e.g., the porous medium equation and the fast diffusion equation, have received a lot of interest in mathematical research and practical applications alike in recent years. Many of them have been interpreted as gradient flows with respect to some metric structure. When it comes to solving partial differential equations of gradient flow type numerically, it is natural to ask for appropriate schemes that preserve the equations' special structure at the discrete level.

In this talk we present a Lagrangian numerical scheme for solving nonlinear degenerate Fokker-Planck equations in multiple space dimensions. The scheme applies to a large class of nonlinear diffusion equations. The key ingredient in our approach is the gradient flow structure of the dynamics. For discretization of the Lagrangian map, we use a finite subspace of linear maps in space and a variational form of the implicit Euler method in time. Thanks to that time discretisation, the fully discrete solution inherits energy estimates from the original gradient flow, and these lead to weak compactness of the trajectories in the continuous limit. We discuss the consistency of the scheme in two space dimensions and present numerical experiments for the porous medium equation.